A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of Rs. 498.96. If the cost of white-washing is 2.00 per square metre, find the
i) inside surface area of the dome,
ii) volume of the air inside the dome.

i)Given :
Cost of white washing the dome from inside = Rs. 498.96
Also, Cost of white washing 1 \({m}^2\) area = Rs. 2.00

Therefore, CSA of the inner side of dome
= Rs. \(\frac{498.96}{2} {m}^2 \)
= \( 249.48 {m}^2\)

ii) Let the inner radius of the hemispherical dome be r.

Now, CSA of inner side of dome = 249.48 \({m}^2\)

But, CSA = \(2{\pi}{r}^2\)
\(\Rightarrow \) 249.48 \({m}^2\) = \(2 × \frac{22}{7} × {r}^2\)
\(\Rightarrow \) \({r}^2 = (\frac{249.48 × 7}{2 × 22})\) \({m}^2\)
\(\Rightarrow \) r = 6.3 m.

Also, we have,

Volume of air inside the dome = Volume of hemispherical dome
= \(\frac{2}{3}{\pi}{r}^3\)
= \(\frac{2}{3} × \frac{22}{7} × {6.3}^3\) \({m}^3\)
= \(523.908\) \({m}^3\) (approx.)

Therefore, volume of the air inside the dome is \(523.908\) \({m}^3\).